Teaching Teaching OverviewFinite difference methods for differential equations, AMCS 252:The course covers theory and algorithms for the numerical solution of ordinary differential equations (ODEs) and of partial differential equations (PDEs) of parabolic, hyperbolic, and elliptic type. Theoretical concepts include:accuracyzero-stabilityabsolute stabilityconvergenceorder of accuracystiffnessconservationCFL condition Algorithms covered include:finite differences steady and unsteady discretization in one and two dimensionsNewton methodsRunge-Kutta methodslinear multistep methodsmultigridimplicit methods for stiff problemscentered and upwind methods for wave equationsdimensional splitting and operator splitting Numerical linear algebra, AMCS 251:Numerical analysis describes the construction and analysis of efficient discrete algorithms to solve continuous problems using large amounts of data. It represents the basis for the field of numerical analysis, and should, therefore, be learned and mastered as early as possible. We cover:fundamentals: computing with matrices, execution time, norms, SVDfactorizations: Cholesky decomposition, QR decomposition, LU decompositionleast-square: normal equation, orthonormal equationerror analysis: error measures, conditioning of a problem, machine precision, stability of an algorithmeigenvalue problem: power iteration, QR, SVDiterative methods: Arnoldi, GMRES, conjugate gradient Summation-by-parts operators for partial differential equations, AMCS 394 - Contemporary topic in numerical analysis:The past decade has seen an explosion in popularity of developing methods with the summation-by-parts (SBP) property. This is because the SBP framework offers a simple, yet powerful methodology for the design and analysis of modern algorithms for the solution of PDEs. The SBP concept was originally developed in the finite difference community with the goal of mimicking finite element energy analysis techniques. In recent years, this simple idea has been exponentially generalized enabling a unifying framework for the stability analysis of many spatial discretizations including finite difference, finite volume, flux reconstruction, and continuous/discontinuous Galerkin finite element methods on structured and unstructured polytope meshes for linear and nonlinear conservation laws on conforming and non-conforming grids. The most important consequence of SBP is that it naturally guides the path to stability and robustness as it mimics continuous stability analysis. The SBP concept provides a strong theoretical framework, that is discretization agnostic, for the analysis of existing schemes and the design of flexible high-order numerical approximations that are robust for complex multi-scale applications. The main topics of the course are:introduction of SBP operators through the stability analysis of model problems equations (advection and advection-diffusion equations)construction of collocated Legendre-Gauss-Lobatto SBP operatorsHadamard formalism and extension of SBP operators to nonlinear PDEsone element discretizations analysis and weak imposition of boundary conditions through the simultaneous-approximation terms techniquemulti-element discretizations and analysis and imposition of weak interface couplingdiscontinuous collocated Galerkin method and entropy stability for the Burgers', compressible Euler and Navier-Stokes equationshp-adaptation linear and nonlinear (entropy) stability analysis of nonconforming interfaces
